The GRBs can be observed up to redshift \(z\sim 10\)  due to the intense explosions in the Universe. An important observational aspect of long GRB is the several correlation between the spectral and intensity properties, which suggests that the GRB measurement can be used as a complementary cosmic probe to the standard candles [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, although the mechanism behind GRBs explosions is not completely known yet. One of the most successful proposals is the Amati relation (\(E_{\rm p}\) -\(E_{\rm iso}\)  relation) [3]}, which relates the rest frame spectral peak energy \(E_{\rm p}\)  and the bolometric isotropic-equivalent radiated energy \(E_{\rm iso}\) . However, the application of GRB observations for cosmology is also affected by the so-called circularity problem [3]}, [7]}, [11]}, which arises from the fact that, given the lack of low redshift measurements of GRBs, the \(E_{\rm p}\) -\(E_{\rm iso}\)  correlation is obtained through assuming a background cosmology. For example, using the standard \(\Lambda \) CDM model to calibrate the GRB measurements, it will inevitably return an overall agreement with the concordance model to make constraints on cosmological parameters of any dark energy framework.
Thus, calibrating the Amati relation is a challenge if one dodges the problem of circularity.
Recently, Amati et al. proposed a model-independent technique to overcome the circularity problem with the Amati relation [16]}, and built up a compilation consisting of 193 GRB data points. To calibrate this relation, they fitted the most recent Hubble data with a B\(\acute{\rm e}\) zier parametric curve, and employed the Hubble data to approximate the cosmic evolution. Then, with the combination of this GRB data set and SNIa JLA compilation, they made observational constraints on the flat \(\Lambda \) CDM model, and obtained that \((\Omega _{\rm m}, \alpha , \beta , {\rm M}_{\rm B})=(0.397\pm _{0.039}^{0.040},0.126\pm _{0.012}^{0.011}, 2.610\pm 0.130,19.090\pm 0.037)\)  at \(2\sigma \)  CL. Here,
\(\Omega _{\rm m}\)  denotes the present
dark matter density parameter. The constraints on the cosmological parameters are obviously different from the results of current observations [17]}, [18]}.
